

ON THE TEXTUAL AND SEMANTIC STRUCTURES OF MAPPING RULE AND MULTIPLICATIVE COMPARE PROBLEMS 

Guershon
Harel Thomas
Post Merlyn
Behr 

In analyzing the propositional structure of multiplicative problems, Nesher (1987) identified and formulated three different categories: Mapping Rule, Multiplicative Compare, and Cartesian Multiplication. In this paper we are interested in the textual and the semantic structures of the first two categories. Mapping Rule. In a Mapping Rule problem there is a mapping rule between the two measure spaces from which the units are derived. For example, in the multiplication (M) problem: 



The phrase "8
books on each shelf" is the mapping rule between the measure spaces
"shelves" and "books."
Nesher characterized the two types of division problems in the Mapping Rule category, partitive and quotitive, as follows. A division problem is partitive if the question is about the string which was the mapping rule in the corresponding multiplication problem, such as in the following division (D) problem: 



A division problem is quotitive if the question is about the string which was an existential description in the corresponding multiplication problem, such as in the following division problem:  


Multiplicative Compare. A Multiplicative Compare problem is one in which a onedirectionalscalarfunction is used to compare between two problem quantities. For example, in the multiplication problem  


The phrase "Ruth has 4 times as many marbles as Dan has" is the onedirectionalscalarfunction between the quantities representing Dan’s set of marbles and Ruth’s set of marbles. Nesher did not characterize partitive and quotitive problems in the Multiplicative Compare category. However, according to Greer's (1985) extension of the type of division problems, a problem is partitive or quotitive, respectively, according to whether the divisor is conceived of as the multiplier or as the multiplicand in the corresponding multiplication problem. If we hold that the numbers 6 and 12 in Problem M_{4} are the divisor and the multiplicand, respectively, then based on Greer’s extension, the following division problems (D_{5} and D_{6}) would be partitive and quotitive, respectively. (As can be seen from Problems M_{1}, D_{2}, and D_{3}, Greer’s extension agrees with Nesher’s characterization of Mapping Rule division problems.) 





Thus, using Nesher propositional terminology, we get that a division problem from the Multiplicative Compare category is partitive if the question is on the string which was an existential description in the corresponding multiplication problem (see, for example, division problems D_{5} with respect to the multiplication problem M_{4}). Similarly, a division problem is quotitive if the question is about the string which was the onedirectionalscalarfunction in the corresponding multiplication problem (see, for example, Problems D_{6} with respect to Problem M_{4}). We will see now that these definitions of partitive and quotitive Multiplicative Compare problems are based on a specific interpretation of the phrase "as many as;" a different interpretation of this phrase would lead to opposite definitions. Consider, for example, Problem D5. The phrase "Ruth has 6 times as many marbles as Dan has" can be interpreted as a unitrateperstatement, i.e., for each marble of Dan, there are 6 marbles of Ruth (see Figure 1), or as a lotperstatement, i.e., for Dan’s set of marbles there are 6 sets of marbles of Ruth, each of which is equivalent to Dan’s set (see Figure 2). 

Figure 1


Figure 2


If the phrase "as many as" is interpreted as a unitrateperstatement, then Problem D. would be conceived of as a quotitive and not as a partitive as was indicated earlier. This is because under this interpretation, to find how many marbles Dan has, one needs to find the number of times the set of 6 marbles is contained in the set of 72 marbles (see Figure 1). On the other hand, if the phrase "as many as" is interpreted as a lotperstatement, the problem situation would suggest that (a) there is one set of marbles belongs to Dan, which against it there are 6 sets of marbles belong to Ruth, each of which is equivalent to Dan’s set, and (b) Ruth has 72 marbles. (See Figure 2.) Thus, to find how many marbles Dan has, one needs to find the number of marbles in each Ruth's set. This situation suggests that Problem D5 is of partitive division type. Applying the same analysis to Problem D6, it will be found that the problem is conceived of as partitive or quotitive according to if the phrase "as many as" is interpreted as a lotperstatement or as a unitrateperstatement, respectively.
RELATIONSHIPS BETWEEN PROBLEM STRUCTURES We indicate that under the lotperstatementinterpretation, partitive (quotitive) Mapping Rule problems have the same textual structure as the quotitive (partitive) Multiplicative Compare problems (see Figure 3): The question in a Mapping Rule partitive problem and in a Multiplicative Compare quotitive problem is about the string which was an association (i.e., either as a mapping rule or as a onedirectional scalar function) between two measure spaces in the corresponding multiplication problem. Similarly, the question in the Mapping Rule quotitive problem and in the Multiplicative Compare partitive problem is about the string which was an existential description in the corresponding multiplication problem. On the other hand, under the unitrateperstatement interpretation, the Mapping Rule partitive and quotitive problems are of the same structure as of the Multiplicative Compare partitive and quotitive problems, respectively (see Figure 3). 

Figure 3


AN EXPERIMENT We will suggest now an experiment to answer empirically the question of whether the phrase "as many as" in division problems from the Multiplicative Compare category is interpreted implicitly by students as a unitrateperstatement or as a lotperstatement. This experiment is part of an instrument developed to assess the inservice teachers’ knowledge of multiplicative structures, which is under way and will be reported at Post, Harel and Behr (in preparation). Items from this experiment include the following example. We gave students two variations of a division problem. In the first variation the problem quantities violate the intuitive partitive model but comfort with the intuitive quotitive model (Fischbein, Deri, Nello, and Marino, 1985). This variation can be achieved, for example, by taking the divisor to be a fractional number and smaller than the dividend. The second variation is a problem in which the quantities comforts with the two intuitive models, which can be achieved, for example, by taking the divisor a whole number and smaller than the dividend. Examples of these variations are Problems D_{7} and D_{8}, respectively. 





Fischbein et al. (1985) and others (Greer, 1985; Greer and Mangan, 1984; Mangan, 1986; Tirosh, Graeber, and Glover 1986; Harel, Post, and Behr, in preparation) found that children and teachers as well select a noncorrect operation when they are presented with problems including numbers that conflict with the rules of the primitive models; students’ performance on problems which comforts with the intuitive models is relatively high. Thus, if the phrase "as many as" is interpreted by the students as a lotperstatement, then, as has been shown earlier, the two variations (D_{7} and D_{8}) would be represented as partitive division problems, and consequently, it would be expected that the students will perform better on the second variation (Problem D_{8}), which does not violate the partitive model, than on the first variation (Problem D_{7}), which does violates the partitive model. On the other hand, if the problem is interpreted as a unitrateperstatement, then the problem (in the two variations) would be represented as a quotitive division, and consequently, it would be expected that the students" performance would be equally high on the two variations, since both problems do not violate the intuitive quotitive models.
CONCLUSIONS From this analysis we see that the interpretation of the phrase "as many as" affect the semantic structure of Multiplicative Compare division problems. The pedagogical value of this analysis is that it points out the need to enrich the cultural and educational experiences which underlie children’s understanding of Multiplicative Compare division problems. Students should be able to move from one interpretation to another in order to construct the problem representation that most incorporates with their knowledge. Our analysis of Missing Value Proportion Problems (Harel and Behr, 1988) and research by many others (Davis, 1984; Greeno, 1983; Behr, Lesh, and Post, 1986) demonstrate the importance of the use of different problem representations during the course of a problem solution. The types of the quantities, discrete or continuous, involved in the problem seem to have an impact on the interpretation of the phrase "as many as," and consequently on the semantic interpretation of the problem as quotitive or partitive. As was shown earlier, an "as many as" phrase which involves discrete quantities can be interpreted either as a unitrateperstatement or as a lotperstatement. On the other hand, if the quantities are continuous, it is more likely that the phrase "as many as" would be interpreted as a lotperstatement, such as in the phrase "a mountain range is 124 times as long as a mural of it." However, further considerations needs to be given to this hypothesis and to the analysis described in this paper.
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This research was supported in part by the National Science Foundation under grant No. 4422968. Any opinions, findings, and conclusions expressed are those of the authors and do not reflect the views of National Science Foundation.  